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The problem of an infinite isotropic elastic space subjected to uniform far-field load and containing an isotropic elastic spherical inhomogeneity with Steigmann–Ogden interface is considered. The interface is treated as a shell of vanishing thickness possessing surface tension as well as membrane and bending stiffnesses. The constitutive and equilibrium equations of the Steigmann–Ogden theory for a spherical surface are written in explicit forms. Closed-form analytical solutions are derived for two cases of loading conditions—the hydrostatic loading and deviatoric loading with vanishing surface tension. The single inhomogeneity-based estimates of the effective properties of macroscopically isotropic materials containing spherical inhomogeneities with Steigmann–Ogden interfaces are presented. It is demonstrated that, in the case of vanishing surface tension, the Steigmann–Ogden model describes a special case of thin and stiff uniform interphase layer.

Abstract. In this paper, an approach to modeling fracture incorporating interfacial mechanics is applied to the example of a curvilinear plane strain crack. The classical Neumann boundary condition is augmented with curvature-dependent surface tension. It is shown that the considered model eliminates the integrable crack-tip stress and strain singularities of order 1/2 present in the classical linear fracture mechanics solutions, and also leads to the sharp crack opening that is consistent with empirical observations. Key words. Fracture mechanics, curvilinear crack, surface tension, complex potentials, singular integrodifferential equations.AMS subject classifications. 74B20, 74G70, 74R101. Introduction. A better understanding of fracture processes is of utmost importance for applications. The standard approach to study the behavior and propagation of fracture is based within the framework of linear elastic fracture mechanics (LEFM). This approach has been successfully applied to countless problems and has a vast literature. However, LEFM contains a well-known internal inconsistency. While it is based on the assumption that the stresses and the strains remain small everywhere in the body, LEFM predicts that the stresses and the strains possess an integrable power singularity of the order 1/2 at the crack tips. Moreover, in the case of an interface crack between two dissimilar materials there is an additional oscillating singularity which results in the non-physical interpenetration of the crack faces near the crack tips.Several models have been proposed to eliminate this internal inconsistency. One of the most common approaches, studied by many authors, is to introduce a two-dimensional cohesive zones or three-dimensional process zones near the crack tips. Despite the obvious advantages of this approach it has shortcomings of both theoretical and practical nature, such as difficulty in specifying physically valid constitutive response functions.Since the fracture occurs within nano-or molecular scale processes, it has been argued that it is impossible to describe the fracture effectively within the context of continuum mechanics. Thus, there is a growing literature dedicated to the modeling of fracture using atomistic and lattice based approaches [1,2,3,5,13,14,24,26]. The accuracy of these methods largely depends on the precise description of intermolecular forces which is difficult to do for liquids and solids [13]. This approach may also present some computational challenges. Several atomto-continuum models have been proposed as well. One of the most extensively studied methods of this type, in the context of finite element method (FEM) approximations to continuum models, is the quasi-continuum method introduced by Tadmor et al in 1996 [27]. Based on an atomistic view of material behavior, its continuum aspect comes from the fact that the FEM is based on energy minimization. A different type of atom-to-continuum modeling is a recently proposed approach by Xiao and Belytschko in [28] which involves the intro...

SummaryA problem of an interface crack between two semi-planes made out of different materials under an action of an in-plane loading of general tensile-shear type is treated in a semi-analytical manner with the help of Dirichlet-to-Neumann mappings. The boundaries of the crack and the interface between semi-planes are subjected to a curvature-dependent surface tension. The resulting system of six singular integrodifferential equations is reduced to the system of three Fredholm equations. It is shown that the introduction of the curvature-dependent surface tension eliminates both classical integrable power singularity of the order 1/2 and an oscillating singularity present in a classical linear elasticity solutions. The numerical results are obtained by solving the original system of singular integro-differential equations by approximating unknown functions with Taylor polynomials.

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